I. Field of the Invention
In general, the present invention relates to auto-correlation and cross-correlation functions, and more particularly to the utilization of auto-correlation and cross-correlation functions in a method for locating a source of a primary signal and in a method of localizing signals. The invention also relates to the field of tomography and the acoustical mapping of boundaries.
II. Description of the Related Art
Passive acoustic localization of calling animals is more accurate when a tomographic estimation is made of the sound speed and wind fields (Helstrom, Statistical Theory of Signal Detection, Pergamon Press, New York, pp. 470, 1968; Watkins et al., Woods Hole Oceanogr. Inst. Tech. Rep., 71-60, Woods Hole, Mass., 1971). Localization and estimation of the environmental fields are made from estimates of the arrival time differences of sound at widely-separated receivers. Cross-correlation is a standard method for optimally estimating the arrival time difference of a signal between a pair of receivers. Under the condition that one signal arrives at each of two receivers, the maximum peak in the cross-correlation function has a signal-to-noise ratio that is about 10 log.sub.10 K decibels greater than that in the received data records, where K is the time-bandwidth product of the animal's call (Helstrom, 1968). Thus cross-correlation may significantly increase the range over which animals may be detected and localized (Helstrom, 1968). However, if there are echoes and reflections which reach the receiver along with the direct path, there are many peaks in the cross-correlation function (FIG. 1). Which peak is chosen if the cross-correlations between the echoes and reflections are similar to or greater than from the direct path? In many terrestrial and shallow water environments with imprecisely known boundaries, the first arrival, which may be nearly straight, may be the only useful path for localization since the geometry of the other paths originating from echoes may be difficult to estimate. Cross-correlation does not tell us which peak to choose. For example, suppose there are three multipaths arriving at each receiver. Each cross-correlation between two receivers will have up to 3.times.3=9 peaks. If five receivers are used for localization, there are four independent cross-correlations that can be formed (Helstrom, 1968). One localization technique requires estimating which of the 9.sup.4 =6561 sets of four arrival time differences is correct (Helstrom, 1968). This is the dilemma I faced five years ago when attempting to localize birds in a North Falmouth forest in Massachusetts. The first arrivals were not always the loudest, and there were many multipath at each receiver. In the few cases where the cross-correlation peak was correctly identified as the difference in the first arrivals, localization could be done and a tomographic estimate of the sound speed and wind field could be made (Watkins et al., 1971). In most other cases, neither localization nor tomographic estimation of the environment was feasible because I could not identify the desired peak in the cross-correlation function.
To address this conundrum, I derive a new method which identifies the cross-correlation peak corresponding to the difference in arrival time between the first arrivals at each receiver in the presence of echoes (Sec. I). Its numerical implementation is efficient. The method does not give up the tremendous gain in signal-to-noise ratio achieved with cross-correlation.
The key to unlocking the multipath problem with cross-correlation is to consider the extra information residing in the reception's auto-correlation functions. They often provide enough information to identify the relative arrival times of all the multipaths at each receiver and between receivers (Sec. II). Estimating these relative arrival times is less efficient than estimating the difference in arrival times of the first arrivals.